\(\renewcommand{\hat}[1]{\widehat{#1}}\)

Shared Qs (u01)


  1. Question

    S-IC.A.1: Understand statistics as a process for making inferences to be made about population parameters based on a random sample from that population.

    Background and future connections

    Below are some important definitions:

    In Statistics, we learn a process for making inferences about population parameters based on a random sample from that population. For a simple parameter, like an average or proportion, the sample statistic will be a best-guess estimate for the corresponding population parameter (for estimating the population’s standard deviation, we will use Bessel’s correction to find a best-guess estimate from the sample). Later, we will also learn how to express the amount of uncertainty in the estimation. (See confidence interval and margin of error to preview how we express uncertainty about the population parameter.)

    For now, let’s practice identifying the sample, statistic, population, and parameter.

    Actual Question

    A park ranger wants to know the average age of all the oak trees in a forest. They randomly select \(20\) oak trees and find an average age of \(100\) years.

    The four concepts are described in random order below.

    1. All the oak trees in the forest
    2. The \(20\) oak trees that the ranger selected
    3. The average age of \(100\) years found in the sample of oak trees
    4. The average age of all the oak trees in the forest

    Match the concept to the description.

    1. Sample =
    2. Statistic =
    3. Population =
    4. Parameter =


    Solution


  2. Question

    S-IC.A.1: Understand statistics as a process for making inferences to be made about population parameters based on a random sample from that population.

    Background and future connections

    Below are some important definitions:

    In Statistics, we learn a process for making inferences about population parameters based on a random sample from that population. For a simple parameter, like an average or proportion, the sample statistic will be a best-guess estimate for the corresponding population parameter (for estimating the population’s standard deviation, we will use Bessel’s correction to find a best-guess estimate from the sample). Later, we will also learn how to express the amount of uncertainty in the estimation. (See confidence interval and margin of error to preview how we express uncertainty about the population parameter.)

    For now, let’s practice identifying the sample, statistic, population, and parameter.

    Actual Question

    A librarian is interested in the average number of pages in all the books in the library’s children’s section. They randomly select \(15\) books from the children’s section and find an average of \(80\) pages per book.

    The four concepts are described in random order below.

    1. All the books in the library’s children’s section
    2. The average of \(80\) pages per book found in the sample
    3. The average number of pages in all the books in the library’s children’s section
    4. The \(15\) books that the librarian selected

    Match the concept to the description.

    1. Sample =
    2. Statistic =
    3. Population =
    4. Parameter =


    Solution


  3. Question

    S-IC.A.1: Understand statistics as a process for making inferences to be made about population parameters based on a random sample from that population.

    Background and future connections

    Below are some important definitions:

    In Statistics, we learn a process for making inferences about population parameters based on a random sample from that population. For a simple parameter, like an average or proportion, the sample statistic will be a best-guess estimate for the corresponding population parameter (for estimating the population’s standard deviation, we will use Bessel’s correction to find a best-guess estimate from the sample). Later, we will also learn how to express the amount of uncertainty in the estimation. (See confidence interval and margin of error to preview how we express uncertainty about the population parameter.)

    For now, let’s practice identifying the sample, statistic, population, and parameter.

    Actual Question

    A scientist wants to know the average size of all the leaves on a tree. They collect \(30\) leaves from the tree and measure their area. The average area of the leaves in their sample is \(10\) square centimeters.

    The four concepts are described in random order below.

    1. The \(30\) leaves that the scientist collected
    2. The average size of all the leaves on the tree
    3. All the leaves on the tree
    4. The average area of \(10\) square centimeters found in the sample of leaves

    Match the concept to the description.

    1. Sample =
    2. Statistic =
    3. Population =
    4. Parameter =


    Solution


  4. Question

    S-IC.A.1: Understand statistics as a process for making inferences to be made about population parameters based on a random sample from that population.

    Background and future connections

    Below are some important definitions:

    In Statistics, we learn a process for making inferences about population parameters based on a random sample from that population. For a simple parameter, like an average or proportion, the sample statistic will be a best-guess estimate for the corresponding population parameter (for estimating the population’s standard deviation, we will use Bessel’s correction to find a best-guess estimate from the sample). Later, we will also learn how to express the amount of uncertainty in the estimation. (See confidence interval and margin of error to preview how we express uncertainty about the population parameter.)

    For now, let’s practice identifying the sample, statistic, population, and parameter.

    Actual Question

    A librarian wants to know what percentage of the available books in the library are overdue. They randomly choose \(50\) books from the shelves and note whether each book is overdue. \(10\) of the \(50\) books are overdue.

    The four concepts are described in random order below.

    1. \(20\%\) of the books in the sample are overdue
    2. The percentage of books in the library that are overdue
    3. The \(50\) books the librarian selected
    4. All the books in the library

    Match the concept to the description.

    1. Sample =
    2. Statistic =
    3. Population =
    4. Parameter =


    Solution


  5. Question

    S-IC.A.1: Understand statistics as a process for making inferences to be made about population parameters based on a random sample from that population.

    Background and future connections

    Below are some important definitions:

    In Statistics, we learn a process for making inferences about population parameters based on a random sample from that population. For a simple parameter, like an average or proportion, the sample statistic will be a best-guess estimate for the corresponding population parameter (for estimating the population’s standard deviation, we will use Bessel’s correction to find a best-guess estimate from the sample). Later, we will also learn how to express the amount of uncertainty in the estimation. (See confidence interval and margin of error to preview how we express uncertainty about the population parameter.)

    For now, let’s practice identifying the sample, statistic, population, and parameter.

    Actual Question

    A teacher randomly selected \(25\) students from their class and found that \(10\) of them had been to Disneyland. They wanted to estimate the percentage of all students in their class who have been to Disneyland.

    The four concepts are described in random order below.

    1. The \(25\) students that the teacher selected
    2. The percentage of all students in the teacher’s class who have been to Disneyland
    3. All the students in the teacher’s class
    4. The \(40\%\) of students in the sample who have been to Disneyland

    Match the concept to the description.

    1. Sample =
    2. Statistic =
    3. Population =
    4. Parameter =


    Solution


  6. Question

    S-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    Background and future connections

    You are expected to differentiate between surveys, experiments, and observational studies. I’ve produced some definitions below.

    Personally, I think there can be overlap between surveys and the other two. For example, you could run an experiment by sending two different surveys out to random people to investigate the way wording influences responses. Don’t worry, I didn’t make any tricky scenarios like that in this question.

    But there is a super important difference between experiments and observational studies. In an observational study, the goal is to find correlations between variables. However, CORRELATION IS NOT CAUSATION! In order to prove causation, an experiment must be run, ideally a doubly-blind randomized controlled experiment. These randomized controlled experiments were essential for bringing medicine from witchcraft to modern, beginning around 1900, which is also around the time that Statistics was developed. In fact, in case it is not clear by now, scientific research and Statistics are deeply intertwined.

    Actual questions

    For each scenario, decide whether the situation describes a survey, an observational study, or an experiment.

    Scenario 1

    A company wanted to test the effectiveness of different marketing campaigns on product sales. They launched separate campaigns targeting different audiences and then measured sales for each campaign. They compared sales figures to determine which campaign was most effective in generating sales.

    Scenario 2

    A hospital wanted to gather feedback from patients about their experience with their medical care. They sent out satisfaction surveys to patients after their hospital visits, asking them to rate their experience with doctors, nurses, and other staff, as well as the hospital’s facilities and overall care.

    Scenario 3

    A meteorologist was interested in studying the formation of hurricanes. They tracked hurricane development using radar and satellites, observing the formation of clouds, the movement of air currents, and the development of storms.

    Scenario 4

    A doctor wanted to compare the effectiveness of different treatments for a specific medical condition. They divided patients with the condition into groups, each receiving a different treatment. They then tracked patients’ symptoms and recovery rates, comparing the outcomes of the groups to determine which treatment was most effective.

    Scenario 5

    A museum was interested in learning about visitor preferences for different exhibits. They conducted exit interviews with visitors, asking them about their favorite exhibits, the time they spent exploring each exhibit, and their suggestions for improving the museum experience.



    Solution


  7. Question

    S-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    Background and future connections

    You are expected to differentiate between surveys, experiments, and observational studies. I’ve produced some definitions below.

    Personally, I think there can be overlap between surveys and the other two. For example, you could run an experiment by sending two different surveys out to random people to investigate the way wording influences responses. Don’t worry, I didn’t make any tricky scenarios like that in this question.

    But there is a super important difference between experiments and observational studies. In an observational study, the goal is to find correlations between variables. However, CORRELATION IS NOT CAUSATION! In order to prove causation, an experiment must be run, ideally a doubly-blind randomized controlled experiment. These randomized controlled experiments were essential for bringing medicine from witchcraft to modern, beginning around 1900, which is also around the time that Statistics was developed. In fact, in case it is not clear by now, scientific research and Statistics are deeply intertwined.

    Actual questions

    For each scenario, decide whether the situation describes a survey, an observational study, or an experiment.

    Scenario 1

    A psychologist wanted to test the effectiveness of a new therapy for treating depression. They divided participants with depression into two groups, one receiving the therapy and the other receiving a standard treatment. They then measured participants’ depression symptoms using a validated scale, comparing the results between the two groups.

    Scenario 2

    A gaming company was curious about player preferences for different game modes. They conducted user surveys, asking players about their favorite game modes, the reasons for their preferences, and their suggestions for improving the game.

    Scenario 3

    A researcher was interested in understanding the interactions between different species of birds in a forest. They spent time observing the birds in their natural habitat, noting their feeding patterns, social interactions, and competition for resources.

    Scenario 4

    An author wanted to understand reader responses to their latest novel. They sent out postcards to a sample of readers, asking them to rate the book’s plot, characters, and overall writing style.

    Scenario 5

    A company wanted to test the effectiveness of different product designs on customer satisfaction. They developed different designs for a product and then offered the product to customers in different groups, each receiving a different design. They then collected customer feedback on each design, comparing the satisfaction ratings to determine which design was most appealing to customers.



    Solution


  8. Question

    S-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    Background and future connections

    You are expected to differentiate between surveys, experiments, and observational studies. I’ve produced some definitions below.

    Personally, I think there can be overlap between surveys and the other two. For example, you could run an experiment by sending two different surveys out to random people to investigate the way wording influences responses. Don’t worry, I didn’t make any tricky scenarios like that in this question.

    But there is a super important difference between experiments and observational studies. In an observational study, the goal is to find correlations between variables. However, CORRELATION IS NOT CAUSATION! In order to prove causation, an experiment must be run, ideally a doubly-blind randomized controlled experiment. These randomized controlled experiments were essential for bringing medicine from witchcraft to modern, beginning around 1900, which is also around the time that Statistics was developed. In fact, in case it is not clear by now, scientific research and Statistics are deeply intertwined.

    Actual questions

    For each scenario, decide whether the situation describes a survey, an observational study, or an experiment.

    Scenario 1

    A doctor wanted to test the effectiveness of a new medication for treating a specific ailment. They divided patients with the ailment into two groups, one receiving the medication and the other receiving a placebo. They tracked patients’ symptoms and recovery rates, comparing the outcomes of the two groups.

    Scenario 2

    An author was interested in understanding reader responses to their latest book. They hosted book clubs and online discussions, allowing readers to share their thoughts and opinions on the book, engage in meaningful conversations, and provide feedback.

    Scenario 3

    A geologist was curious about the rock formations in a particular canyon. They spent time exploring the canyon, examining the rocks, noting their composition, and studying the geological processes that formed the canyon.

    Scenario 4

    A company wanted to see if a new advertising campaign would increase sales. They measured sales of a product before launching the campaign and then measured sales again after the campaign was implemented. They compared sales figures before and after the campaign to determine its impact.

    Scenario 5

    A scientist was curious about the effects of different types of light on plant growth. They exposed plants to different wavelengths of light, observing their growth patterns, leaf size, and overall health.



    Solution


  9. Question

    S-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    Background and future connections

    You are expected to differentiate between surveys, experiments, and observational studies. I’ve produced some definitions below.

    Personally, I think there can be overlap between surveys and the other two. For example, you could run an experiment by sending two different surveys out to random people to investigate the way wording influences responses. Don’t worry, I didn’t make any tricky scenarios like that in this question.

    But there is a super important difference between experiments and observational studies. In an observational study, the goal is to find correlations between variables. However, CORRELATION IS NOT CAUSATION! In order to prove causation, an experiment must be run, ideally a doubly-blind randomized controlled experiment. These randomized controlled experiments were essential for bringing medicine from witchcraft to modern, beginning around 1900, which is also around the time that Statistics was developed. In fact, in case it is not clear by now, scientific research and Statistics are deeply intertwined.

    Actual questions

    For each scenario, decide whether the situation describes a survey, an observational study, or an experiment.

    Scenario 1

    A researcher wanted to study the impact of different types of music on memory. They divided participants into three groups, each listening to a different genre of music while memorizing a list of words. They then tested the participants’ recall, comparing their memory performance across the three groups.

    Scenario 2

    A wildlife biologist wanted to study the behavior of a group of wolves in their natural habitat. They spent weeks observing the wolves, noting their hunting patterns, social interactions, and communication signals.

    Scenario 3

    A company wanted to test the effectiveness of different marketing campaigns on product sales. They launched separate campaigns targeting different audiences and then measured sales for each campaign. They compared sales figures to determine which campaign was most effective in generating sales.

    Scenario 4

    A software developer wanted to get feedback from users on their new software update. They sent out emails to a sample of users, asking them to complete a questionnaire about the update’s functionality, usability, and overall performance.

    Scenario 5

    A financial advisor was interested in understanding client investment goals and risk tolerance. They conducted online surveys, asking clients about their financial objectives, investment horizons, and comfort levels with different investment strategies.



    Solution


  10. Question

    S-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    Background and future connections

    You are expected to differentiate between surveys, experiments, and observational studies. I’ve produced some definitions below.

    Personally, I think there can be overlap between surveys and the other two. For example, you could run an experiment by sending two different surveys out to random people to investigate the way wording influences responses. Don’t worry, I didn’t make any tricky scenarios like that in this question.

    But there is a super important difference between experiments and observational studies. In an observational study, the goal is to find correlations between variables. However, CORRELATION IS NOT CAUSATION! In order to prove causation, an experiment must be run, ideally a doubly-blind randomized controlled experiment. These randomized controlled experiments were essential for bringing medicine from witchcraft to modern, beginning around 1900, which is also around the time that Statistics was developed. In fact, in case it is not clear by now, scientific research and Statistics are deeply intertwined.

    Actual questions

    For each scenario, decide whether the situation describes a survey, an observational study, or an experiment.

    Scenario 1

    A researcher wanted to study the effects of different levels of exercise on cognitive function. They divided participants into groups, each engaging in a different level of exercise. They then tested participants’ cognitive abilities using a standardized test, comparing the results between the groups to determine the impact of exercise on cognitive function.

    Scenario 2

    A wildlife biologist was interested in studying the migration patterns of a species of bird. They tracked the birds’ movements using GPS tags, analyzing their flight paths, stopover locations, and environmental factors influencing their migration.

    Scenario 3

    An oceanographer wanted to study the impact of pollution on marine life. They collected water samples from different locations in the ocean, analyzing their chemical composition and observing the presence of pollutants and their effects on marine organisms.

    Scenario 4

    A news organization wanted to understand how readers reacted to a particular news story. They conducted focus groups with readers, discussing their reactions to the story, their understanding of the information presented, and their opinions on the reporting.

    Scenario 5

    A technology company wanted to gather feedback from users on their new mobile app. They sent out emails to a sample of users, asking them to complete a questionnaire about their experience with the app.



    Solution


  11. Question

    S-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    When choosing who to survey, test, or observe, it is important the sample is representative of the population. The ideal method is a simple random sample, but sometimes a true simple random sample is difficult to achieve in practice. Other methods can introduce selection bias, where the population is unevenly sampled, so the sample is not representatitve of the population.

    In each of the scenarios below, identify which sampling method would be more biased and which is closer to a simple random sample (unbiased).

    Scenario 1

    A chef wants to know the average number of customers who order appetizers.

    sample method 1 sample method 2
    The chef writes down the time of each order on a slip of paper, puts the papers in a hat, and has a server draw five slips at random, then checks the orders from those times. The chef checks orders from customers who have ordered the most expensive main courses.

    Scenario 2

    A dog groomer wants to know the average weight of dogs they groom.

    sample method 1 sample method 2
    The groomer uses a deck of cards to randomly select five appointments from their schedule, ensuring each dog has an equal chance of being chosen. The groomer weighs dogs that are scheduled for specific types of grooming services.

    Scenario 3

    A researcher wants to know the average number of hours people spend exercising each week.

    sample method 1 sample method 2
    The researcher uses a phone directory and selects 10 phone numbers at random, ensuring each person has an equal chance of being chosen. The researcher surveys people they meet at a local gym.


    Solution


  12. Question

    S-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    When choosing who to survey, test, or observe, it is important the sample is representative of the population. The ideal method is a simple random sample, but sometimes a true simple random sample is difficult to achieve in practice. Other methods can introduce selection bias, where the population is unevenly sampled, so the sample is not representatitve of the population.

    In each of the scenarios below, identify which sampling method would be more biased and which is closer to a simple random sample (unbiased).

    Scenario 1

    A museum curator wants to know the average number of visitors to the museum each day.

    sample method 1 sample method 2
    The curator counts visitors who enter the museum during a special exhibition. The curator uses a random number generator to select five hours of the day, ensuring each hour has an equal chance of being chosen, and then counts the visitors during those hours.

    Scenario 2

    A gardener wants to know the average number of buds on each flower.

    sample method 1 sample method 2
    The gardener counts buds on flowers located in a specific area of the garden that receives more water. The gardener closes their eyes and walks randomly through the garden, counting the buds on the first 10 flowers they encounter.

    Scenario 3

    A pet sitter wants to know the average number of hours they spend walking dogs each day.

    sample method 1 sample method 2
    The pet sitter writes each dog’s name on a separate piece of paper, puts the papers in a bowl, and has a child draw three names at random. The pet sitter checks the time spent walking dogs that are large breeds.


    Solution


  13. Question

    S-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    When choosing who to survey, test, or observe, it is important the sample is representative of the population. The ideal method is a simple random sample, but sometimes a true simple random sample is difficult to achieve in practice. Other methods can introduce selection bias, where the population is unevenly sampled, so the sample is not representatitve of the population.

    In each of the scenarios below, identify which sampling method would be more biased and which is closer to a simple random sample (unbiased).

    Scenario 1

    A teacher wants to know the average number of hours students spend working on their projects.

    sample method 1 sample method 2
    The teacher asks students who have already finished their projects. The teacher writes each student’s name on a separate piece of paper, puts the papers in a bowl, and has a student draw five names at random.

    Scenario 2

    A salesperson wants to know the average number of sales they make each week.

    sample method 1 sample method 2
    The salesperson writes each day of the week on a separate piece of paper, puts the papers in a bowl, and has a colleague draw three names at random. The salesperson checks the sales records for days when they are actively involved in a sales promotion.

    Scenario 3

    A store manager wants to know the average number of customers who return items.

    sample method 1 sample method 2
    The manager uses a random number generator to select 10 receipts from the past month, ensuring each receipt has an equal chance of being chosen. The manager checks the returns from customers who made purchases on a specific holiday.


    Solution


  14. Question

    S-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    When choosing who to survey, test, or observe, it is important the sample is representative of the population. The ideal method is a simple random sample, but sometimes a true simple random sample is difficult to achieve in practice. Other methods can introduce selection bias, where the population is unevenly sampled, so the sample is not representatitve of the population.

    In each of the scenarios below, identify which sampling method would be more biased and which is closer to a simple random sample (unbiased).

    Scenario 1

    A musician wants to know the average number of songs they write each year.

    sample method 1 sample method 2
    The musician checks their songwriting records for years when they were actively touring. The musician writes each year on a separate piece of paper, puts the papers in a bowl, and has a friend draw three names at random.

    Scenario 2

    A teacher wants to know the average number of minutes students spend completing their projects.

    sample method 1 sample method 2
    The teacher writes each student’s name on a separate piece of paper, puts the papers in a bowl, and has a student draw five names at random. The teacher asks students who have already finished their projects.

    Scenario 3

    A researcher wants to know the average number of hours people spend commuting each day.

    sample method 1 sample method 2
    The researcher surveys people they meet on public transportation. The researcher uses a phone directory and selects 10 phone numbers at random, ensuring each person has an equal chance of being chosen.


    Solution


  15. Question

    S-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    When choosing who to survey, test, or observe, it is important the sample is representative of the population. The ideal method is a simple random sample, but sometimes a true simple random sample is difficult to achieve in practice. Other methods can introduce selection bias, where the population is unevenly sampled, so the sample is not representatitve of the population.

    In each of the scenarios below, identify which sampling method would be more biased and which is closer to a simple random sample (unbiased).

    Scenario 1

    A musician wants to know the average number of concerts they play each month.

    sample method 1 sample method 2
    The musician uses a random number generator to select three months from the past year, ensuring each month has an equal chance of being chosen. The musician checks their concert schedule for months when they were touring in specific regions.

    Scenario 2

    A teacher wants to know the average number of hours students spend working on their homework.

    sample method 1 sample method 2
    The teacher writes each student’s name on a separate piece of paper, puts the papers in a bowl, and has a student draw five names at random. The teacher asks students who have completed their homework early.

    Scenario 3

    A store owner wants to know the average number of customers who use a discount code.

    sample method 1 sample method 2
    The owner checks the discount code usage for customers who have made purchases during specific promotions. The owner picks a day at random from the past week and examines every receipt from that day, ensuring each customer has an equal chance of being included.


    Solution


  16. Question

    S-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    There is a key difference between experiments and observational studies. In an experiment, the scientist actively engages with one group differently than another. Ideally, only one small element is altered between the groups, to isolate the effects of changing the independent variable. This allows an experiment to conclude that the change in the independent variable caused the difference in outcomes.

    In an observational study, the researcher merely measures variables, without actively changing one variable. Because of this, an observational study merely establishes a correlation.

    In each scenario below, a research question is presented. Two methodologies are shown. Decide which methodology could lead to a cause-effect conclusion, and which methodology would be limited to showing a correlation.

    Scenario 1

    Does listening to music while studying improve memory retention?

    study design 1 study design 2
    study: The researcher divides 100 students into two groups using a random number generator: one group listens to music, the other studies in silence. Then, the researcher tests their memory retention of a specific set of information. The researcher observes the memory performance of 50 students who listen to music while studying and 50 students who study in silence.
    type of conclusion:

    Scenario 2

    Does drinking coffee improve reaction time?

    study design 1 study design 2
    study: The researcher examines the average reaction times of 100 people who drink coffee regularly and 100 people who do not drink coffee regularly. The researcher puts 200 participants into groups by flipping a coin: one group drinks coffee, the other gets a placebo. Then, the researcher measures their reaction times in a standardized task.
    type of conclusion:

    Scenario 3

    Does consuming a specific type of yogurt improve gut health?

    study design 1 study design 2
    study: A researcher divides 100 people into groups by spinning a spinner: one group consumes the specific yogurt, the other consumes a placebo. Then, the researcher measures their gut health using standardized tests. The researcher investigates the gut health of 50 people who consume the specific yogurt and 50 people who do not consume any yogurt.
    type of conclusion:


    Solution


  17. Question

    S-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    There is a key difference between experiments and observational studies. In an experiment, the scientist actively engages with one group differently than another. Ideally, only one small element is altered between the groups, to isolate the effects of changing the independent variable. This allows an experiment to conclude that the change in the independent variable caused the difference in outcomes.

    In an observational study, the researcher merely measures variables, without actively changing one variable. Because of this, an observational study merely establishes a correlation.

    In each scenario below, a research question is presented. Two methodologies are shown. Decide which methodology could lead to a cause-effect conclusion, and which methodology would be limited to showing a correlation.

    Scenario 1

    Does wearing orange clothing improve energy levels?

    study design 1 study design 2
    study: The researcher looks at the energy levels of 50 people wearing orange clothing and 50 people wearing other colors. The researcher assigns 100 participants to groups by drawing cards from a deck: one group wears orange clothing, the other wears a different color. Then, the researcher measures their energy levels using a standardized questionnaire.
    type of conclusion:

    Scenario 2

    Does consuming a specific type of dark chocolate improve cognitive function?

    study design 1 study design 2
    study: The researcher investigates the cognitive function of 50 people who consume the specific dark chocolate and 50 people who do not consume any dark chocolate. A researcher divides 100 people into groups by spinning a spinner: one group consumes the specific dark chocolate, the other consumes a placebo. Then, the researcher measures their cognitive function using standardized tests.
    type of conclusion:

    Scenario 3

    Does spending time outdoors improve cognitive function?

    study design 1 study design 2
    study: The researcher divides 100 participants into two groups using a random number generator: one group spends time outdoors, the other stays indoors. Then, the researcher measures their cognitive function using standardized tests. The researcher compares the cognitive performance of 50 people who spend time outdoors regularly to the cognitive performance of 50 people who do not spend time outdoors regularly.
    type of conclusion:


    Solution


  18. Question

    S-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    There is a key difference between experiments and observational studies. In an experiment, the scientist actively engages with one group differently than another. Ideally, only one small element is altered between the groups, to isolate the effects of changing the independent variable. This allows an experiment to conclude that the change in the independent variable caused the difference in outcomes.

    In an observational study, the researcher merely measures variables, without actively changing one variable. Because of this, an observational study merely establishes a correlation.

    In each scenario below, a research question is presented. Two methodologies are shown. Decide which methodology could lead to a cause-effect conclusion, and which methodology would be limited to showing a correlation.

    Scenario 1

    Does listening to nature sounds improve sleep quality?

    study design 1 study design 2
    study: A researcher divides 100 students into two groups using a random number generator: one group listens to nature sounds, the other listens to white noise. Then, the researcher compares their sleep quality scores. The researcher observes the sleep quality of 50 people who listen to nature sounds and 50 people who listen to white noise.
    type of conclusion:

    Scenario 2

    Does using a specific type of breathing exercise reduce stress?

    study design 1 study design 2
    study: The researcher compares the stress levels of 50 people who use the specific breathing exercise to the stress levels of 50 people who do not use any breathing exercises. The researcher divides 100 participants into two groups by flipping a coin: one group uses the specific breathing exercise, the other uses a placebo. Then, the researcher measures their stress levels using a standardized questionnaire.
    type of conclusion:

    Scenario 3

    Does spending time outdoors improve cognitive function?

    study design 1 study design 2
    study: The researcher compares the cognitive performance of 50 people who spend time outdoors regularly to the cognitive performance of 50 people who do not spend time outdoors regularly. The researcher divides 100 participants into two groups using a random number generator: one group spends time outdoors, the other stays indoors. Then, the researcher measures their cognitive function using standardized tests.
    type of conclusion:


    Solution


  19. Question

    S-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    There is a key difference between experiments and observational studies. In an experiment, the scientist actively engages with one group differently than another. Ideally, only one small element is altered between the groups, to isolate the effects of changing the independent variable. This allows an experiment to conclude that the change in the independent variable caused the difference in outcomes.

    In an observational study, the researcher merely measures variables, without actively changing one variable. Because of this, an observational study merely establishes a correlation.

    In each scenario below, a research question is presented. Two methodologies are shown. Decide which methodology could lead to a cause-effect conclusion, and which methodology would be limited to showing a correlation.

    Scenario 1

    Does taking turmeric supplements improve joint health?

    study design 1 study design 2
    study: The researcher puts 200 participants into groups by drawing straws: one group takes turmeric supplements, the other gets a placebo. Then, the researcher monitors their joint health over a long period. The researcher examines the joint health of 100 people who take turmeric supplements and 100 people who do not take turmeric supplements.
    type of conclusion:

    Scenario 2

    Does using a specific type of essential oil reduce headaches?

    study design 1 study design 2
    study: A researcher places 100 people into groups by drawing straws: one group uses the specific essential oil, the other uses a placebo. Then, the researcher records their headache frequency over a set period. The researcher examines the frequency of headaches in 50 people who use the specific essential oil and 50 people who do not use any essential oils.
    type of conclusion:

    Scenario 3

    Does consuming a specific type of yogurt improve gut health?

    study design 1 study design 2
    study: The researcher investigates the gut health of 50 people who consume the specific yogurt and 50 people who do not consume any yogurt. A researcher divides 100 people into groups by spinning a spinner: one group consumes the specific yogurt, the other consumes a placebo. Then, the researcher measures their gut health using standardized tests.
    type of conclusion:


    Solution


  20. Question

    S-IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    There is a key difference between experiments and observational studies. In an experiment, the scientist actively engages with one group differently than another. Ideally, only one small element is altered between the groups, to isolate the effects of changing the independent variable. This allows an experiment to conclude that the change in the independent variable caused the difference in outcomes.

    In an observational study, the researcher merely measures variables, without actively changing one variable. Because of this, an observational study merely establishes a correlation.

    In each scenario below, a research question is presented. Two methodologies are shown. Decide which methodology could lead to a cause-effect conclusion, and which methodology would be limited to showing a correlation.

    Scenario 1

    Does consuming a specific type of probiotic improve gut health?

    study design 1 study design 2
    study: The researcher investigates the gut health of 50 people who consume the specific probiotic and 50 people who do not consume any probiotics. A researcher divides 100 people into groups by spinning a spinner: one group consumes the specific probiotic, the other consumes a placebo. Then, the researcher measures their gut health using standardized tests.
    type of conclusion:

    Scenario 2

    Does using a specific type of yoga routine improve flexibility?

    study design 1 study design 2
    study: A researcher places 100 people into groups by drawing straws: one group does the specific yoga routine, the other does not practice yoga. Then, the researcher measures their flexibility after a set period. The researcher examines the flexibility of 50 people who do the specific yoga routine and 50 people who do not practice yoga regularly.
    type of conclusion:

    Scenario 3

    Does using a specific type of aromatherapy oil reduce stress?

    study design 1 study design 2
    study: A researcher places 100 people into groups by drawing straws: one group uses the specific aromatherapy oil, the other uses a different oil. Then, the researcher measures their stress levels after a set period. The researcher examines the stress levels of 50 people who use the specific aromatherapy oil and 50 people who use a different oil.
    type of conclusion:


    Solution


  21. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A spinner was made to model a discrete-uniform distribution.

    plot of chunk unnamed-chunk-1

    The spinner was spun a few times, producing the values below.

    x
    42
    42
    46
    53
    42
    58
    51
    56
    48
    57
    56
    57


    Please help describe the values by finding a few statistics.

    The sample size, \(n\)

    The sample size is simply a count of how many values are in the sample.

    The sample total, \(\sum x\)

    The sample total is found by adding up all the values. Informally, we write it as \(\sum x\). More formally, we write it as \(\sum\limits_{i=1}^n x_i\). Notice, both methods refer to the summation operator.

    The sample mean, \(\bar{x}\)

    The sample mean, or in common parlance “the sample average”, is found by dividing the sample total by the sample size. The symbol \(\bar{x}\) is pronounced “x bar”. Please round to the nearest tenth. \[\bar{x} = \frac{\sum x}{n}\]

    The sample median

    The median represents a boundary between the lower half of the values and the upper half of the values. This should be review. Remember, it helps to sort the data. We could write a formula using a sort function and floor and ceiling functions:

    \[\mathrm{median}(x) ~=~ \frac{\mathrm{sort}(x)_{\lceil n/2\rceil}+\mathrm{sort}(x)_{\lfloor n/2+1\rfloor}}{2}\]

    But, you’ll probably just want to sort the values. Then, if the sample size is odd, find the middle value. If the sample size is even, find the mean of the two middle values.

    A conditional count

    In data analysis, we often use conditional counts. In this case, I want you to count how many values are greater than 56. I am using the # symbol to represent a count, and in the parentheses I am denoting a condition (with an inequality).

    A sample proportion

    What proportion of the values are less than 48? A proportion is a conditional count divided by the sample size. Please round to the nearest hundredth.



    Solution


  22. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A spinner was made to model a discrete-uniform distribution.

    plot of chunk unnamed-chunk-1

    The spinner was spun a few times, producing the values below.

    x
    51
    46
    48
    52
    41
    41
    58
    56
    41
    50
    42


    Please help describe the values by finding a few statistics.

    The sample size, \(n\)

    The sample size is simply a count of how many values are in the sample.

    The sample total, \(\sum x\)

    The sample total is found by adding up all the values. Informally, we write it as \(\sum x\). More formally, we write it as \(\sum\limits_{i=1}^n x_i\). Notice, both methods refer to the summation operator.

    The sample mean, \(\bar{x}\)

    The sample mean, or in common parlance “the sample average”, is found by dividing the sample total by the sample size. The symbol \(\bar{x}\) is pronounced “x bar”. Please round to the nearest tenth. \[\bar{x} = \frac{\sum x}{n}\]

    The sample median

    The median represents a boundary between the lower half of the values and the upper half of the values. This should be review. Remember, it helps to sort the data. We could write a formula using a sort function and floor and ceiling functions:

    \[\mathrm{median}(x) ~=~ \frac{\mathrm{sort}(x)_{\lceil n/2\rceil}+\mathrm{sort}(x)_{\lfloor n/2+1\rfloor}}{2}\]

    But, you’ll probably just want to sort the values. Then, if the sample size is odd, find the middle value. If the sample size is even, find the mean of the two middle values.

    A conditional count

    In data analysis, we often use conditional counts. In this case, I want you to count how many values are greater than 43. I am using the # symbol to represent a count, and in the parentheses I am denoting a condition (with an inequality).

    A sample proportion

    What proportion of the values are less than 54? A proportion is a conditional count divided by the sample size. Please round to the nearest hundredth.



    Solution


  23. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A spinner was made to model a discrete-uniform distribution.

    plot of chunk unnamed-chunk-1

    The spinner was spun a few times, producing the values below.

    x
    44
    47
    46
    59
    43
    56
    49
    43
    48
    48
    54


    Please help describe the values by finding a few statistics.

    The sample size, \(n\)

    The sample size is simply a count of how many values are in the sample.

    The sample total, \(\sum x\)

    The sample total is found by adding up all the values. Informally, we write it as \(\sum x\). More formally, we write it as \(\sum\limits_{i=1}^n x_i\). Notice, both methods refer to the summation operator.

    The sample mean, \(\bar{x}\)

    The sample mean, or in common parlance “the sample average”, is found by dividing the sample total by the sample size. The symbol \(\bar{x}\) is pronounced “x bar”. Please round to the nearest tenth. \[\bar{x} = \frac{\sum x}{n}\]

    The sample median

    The median represents a boundary between the lower half of the values and the upper half of the values. This should be review. Remember, it helps to sort the data. We could write a formula using a sort function and floor and ceiling functions:

    \[\mathrm{median}(x) ~=~ \frac{\mathrm{sort}(x)_{\lceil n/2\rceil}+\mathrm{sort}(x)_{\lfloor n/2+1\rfloor}}{2}\]

    But, you’ll probably just want to sort the values. Then, if the sample size is odd, find the middle value. If the sample size is even, find the mean of the two middle values.

    A conditional count

    In data analysis, we often use conditional counts. In this case, I want you to count how many values are at least 48. I am using the # symbol to represent a count, and in the parentheses I am denoting a condition (with an inequality).

    A sample proportion

    What proportion of the values are greater than 48? A proportion is a conditional count divided by the sample size. Please round to the nearest hundredth.



    Solution


  24. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A spinner was made to model a discrete-uniform distribution.

    plot of chunk unnamed-chunk-1

    The spinner was spun a few times, producing the values below.

    x
    31
    55
    46
    30
    57
    33
    50
    50
    51
    54
    53


    Please help describe the values by finding a few statistics.

    The sample size, \(n\)

    The sample size is simply a count of how many values are in the sample.

    The sample total, \(\sum x\)

    The sample total is found by adding up all the values. Informally, we write it as \(\sum x\). More formally, we write it as \(\sum\limits_{i=1}^n x_i\). Notice, both methods refer to the summation operator.

    The sample mean, \(\bar{x}\)

    The sample mean, or in common parlance “the sample average”, is found by dividing the sample total by the sample size. The symbol \(\bar{x}\) is pronounced “x bar”. Please round to the nearest tenth. \[\bar{x} = \frac{\sum x}{n}\]

    The sample median

    The median represents a boundary between the lower half of the values and the upper half of the values. This should be review. Remember, it helps to sort the data. We could write a formula using a sort function and floor and ceiling functions:

    \[\mathrm{median}(x) ~=~ \frac{\mathrm{sort}(x)_{\lceil n/2\rceil}+\mathrm{sort}(x)_{\lfloor n/2+1\rfloor}}{2}\]

    But, you’ll probably just want to sort the values. Then, if the sample size is odd, find the middle value. If the sample size is even, find the mean of the two middle values.

    A conditional count

    In data analysis, we often use conditional counts. In this case, I want you to count how many values are greater than 34. I am using the # symbol to represent a count, and in the parentheses I am denoting a condition (with an inequality).

    A sample proportion

    What proportion of the values are at most 41? A proportion is a conditional count divided by the sample size. Please round to the nearest hundredth.



    Solution


  25. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A spinner was made to model a discrete-uniform distribution.

    plot of chunk unnamed-chunk-1

    The spinner was spun a few times, producing the values below.

    x
    46
    66
    48
    59
    73
    76
    68
    50
    41
    42
    77
    80


    Please help describe the values by finding a few statistics.

    The sample size, \(n\)

    The sample size is simply a count of how many values are in the sample.

    The sample total, \(\sum x\)

    The sample total is found by adding up all the values. Informally, we write it as \(\sum x\). More formally, we write it as \(\sum\limits_{i=1}^n x_i\). Notice, both methods refer to the summation operator.

    The sample mean, \(\bar{x}\)

    The sample mean, or in common parlance “the sample average”, is found by dividing the sample total by the sample size. The symbol \(\bar{x}\) is pronounced “x bar”. Please round to the nearest tenth. \[\bar{x} = \frac{\sum x}{n}\]

    The sample median

    The median represents a boundary between the lower half of the values and the upper half of the values. This should be review. Remember, it helps to sort the data. We could write a formula using a sort function and floor and ceiling functions:

    \[\mathrm{median}(x) ~=~ \frac{\mathrm{sort}(x)_{\lceil n/2\rceil}+\mathrm{sort}(x)_{\lfloor n/2+1\rfloor}}{2}\]

    But, you’ll probably just want to sort the values. Then, if the sample size is odd, find the middle value. If the sample size is even, find the mean of the two middle values.

    A conditional count

    In data analysis, we often use conditional counts. In this case, I want you to count how many values are greater than 55. I am using the # symbol to represent a count, and in the parentheses I am denoting a condition (with an inequality).

    A sample proportion

    What proportion of the values are less than 68? A proportion is a conditional count divided by the sample size. Please round to the nearest hundredth.



    Solution


  26. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A spinner was made to model a discrete-uniform distribution.

    plot of chunk unnamed-chunk-1

    The spinner was spun many times, producing the values below.

    x
    1 12 12 8 2
    10 11 2 10 1
    8 6 14 7 3
    5 3 1 1 3
    8 4 11 5 5
    5 9 3 1 12
    6 11 14 7 14
    14 12 12 6 3
    14 14 12 9 9
    1 8 8 5 9
    6 13 8 10 12
    5 5 11 11 3
    10 7 9 7 4


    Please help describe the values by finding a few statistics.

    The sample size, \(n\)

    The sample size is simply a count of how many values are in the sample.

    The sample total, \(\sum x\)

    The sample total is found by adding up all the values. Informally, we write it as \(\sum x\). More formally, we write it as \(\sum\limits_{i=1}^n x_i\). Notice, both methods refer to the summation operator.

    The sample mean, \(\bar{x}\)

    The sample mean, or in common parlance “the sample average”, is found by dividing the sample total by the sample size. The symbol \(\bar{x}\) is pronounced “x bar”. Please round to the nearest tenth. \[\bar{x} = \frac{\sum x}{n}\]

    The sample median

    The median represents a boundary between the lower half of the values and the upper half of the values. This should be review. Remember, it helps to sort the data. We could write a formula using a sort function and floor and ceiling functions:

    \[\mathrm{median}(x) ~=~ \frac{\mathrm{sort}(x)_{\lceil n/2\rceil}+\mathrm{sort}(x)_{\lfloor n/2+1\rfloor}}{2}\]

    But, you’ll probably just want to sort the values. Then, if the sample size is odd, find the middle value. If the sample size is even, find the mean of the two middle values.

    A conditional count

    In data analysis, we often use conditional counts. In this case, I want you to count how many values are greater than 7. I am using the # symbol to represent a count, and in the parentheses I am denoting a condition (with an inequality).

    A sample proportion

    What proportion of the values are less than 6? A proportion is a conditional count divided by the sample size. Please round to the nearest hundredth.



    Solution


  27. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A spinner was made to model a discrete-uniform distribution.

    plot of chunk unnamed-chunk-1

    The spinner was spun many times, producing the values below.

    x
    7 2 23 20 5
    8 14 6 13 22
    15 16 22 10 23
    21 9 17 16 11
    19 1 2 21 6
    2 20 15 9 8
    1 15 20 10 24
    11 9 24 21 7
    1 23 10 16 4
    18 17 24 20 13
    18 23 14 10 20
    7 1 19 18 20
    22 1 19 15 18
    23 20 13 8 12
    11 19 10 17 3
    12 1 22 4 16
    2 7 20 18 10


    Please help describe the values by finding a few statistics.

    The sample size, \(n\)

    The sample size is simply a count of how many values are in the sample.

    The sample total, \(\sum x\)

    The sample total is found by adding up all the values. Informally, we write it as \(\sum x\). More formally, we write it as \(\sum\limits_{i=1}^n x_i\). Notice, both methods refer to the summation operator.

    The sample mean, \(\bar{x}\)

    The sample mean, or in common parlance “the sample average”, is found by dividing the sample total by the sample size. The symbol \(\bar{x}\) is pronounced “x bar”. Please round to the nearest tenth. \[\bar{x} = \frac{\sum x}{n}\]

    The sample median

    The median represents a boundary between the lower half of the values and the upper half of the values. This should be review. Remember, it helps to sort the data. We could write a formula using a sort function and floor and ceiling functions:

    \[\mathrm{median}(x) ~=~ \frac{\mathrm{sort}(x)_{\lceil n/2\rceil}+\mathrm{sort}(x)_{\lfloor n/2+1\rfloor}}{2}\]

    But, you’ll probably just want to sort the values. Then, if the sample size is odd, find the middle value. If the sample size is even, find the mean of the two middle values.

    A conditional count

    In data analysis, we often use conditional counts. In this case, I want you to count how many values are less than 11. I am using the # symbol to represent a count, and in the parentheses I am denoting a condition (with an inequality).

    A sample proportion

    What proportion of the values are at least 12? A proportion is a conditional count divided by the sample size. Please round to the nearest hundredth.



    Solution


  28. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A spinner was made to model a discrete-uniform distribution.

    plot of chunk unnamed-chunk-1

    The spinner was spun many times, producing the values below.

    x
    14 22 18 16 18 13 3 16 14
    19 22 16 1 13 14 10 4 15
    10 11 18 16 14 6 16 22 21
    7 15 19 17 15 13 22 15 21
    21 17 12 19 18 18 20 7 17
    21 21 5 8 21 15 5 11 20
    2 2 20 22 10 7 19 9 13
    22 10 11 3 14 22 5 11 1
    3 2 5 22 20 11 21 4 5
    10 17 1 5 19 22 2 3 12
    4 7 8 2 1 3 6 6 19
    2 19 1 16 16 18 22 6 13
    20 5 15 19 1 15 2 18 8


    Please help describe the values by finding a few statistics.

    The sample size, \(n\)

    The sample size is simply a count of how many values are in the sample.

    The sample total, \(\sum x\)

    The sample total is found by adding up all the values. Informally, we write it as \(\sum x\). More formally, we write it as \(\sum\limits_{i=1}^n x_i\). Notice, both methods refer to the summation operator.

    The sample mean, \(\bar{x}\)

    The sample mean, or in common parlance “the sample average”, is found by dividing the sample total by the sample size. The symbol \(\bar{x}\) is pronounced “x bar”. Please round to the nearest tenth. \[\bar{x} = \frac{\sum x}{n}\]

    The sample median

    The median represents a boundary between the lower half of the values and the upper half of the values. This should be review. Remember, it helps to sort the data. We could write a formula using a sort function and floor and ceiling functions:

    \[\mathrm{median}(x) ~=~ \frac{\mathrm{sort}(x)_{\lceil n/2\rceil}+\mathrm{sort}(x)_{\lfloor n/2+1\rfloor}}{2}\]

    But, you’ll probably just want to sort the values. Then, if the sample size is odd, find the middle value. If the sample size is even, find the mean of the two middle values.

    A conditional count

    In data analysis, we often use conditional counts. In this case, I want you to count how many values are at least 15. I am using the # symbol to represent a count, and in the parentheses I am denoting a condition (with an inequality).

    A sample proportion

    What proportion of the values are at most 18? A proportion is a conditional count divided by the sample size. Please round to the nearest hundredth.



    Solution


  29. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A spinner was made to model a discrete-uniform distribution.

    plot of chunk unnamed-chunk-1

    The spinner was spun many times, producing the values below.

    x
    13 11 15 15 8
    2 16 11 11 10
    1 4 8 14 6
    13 4 10 4 16
    6 3 7 6 4
    2 14 2 6 3
    9 15 7 1 7
    15 11 5 11 8
    1 6 7 7 8
    2 8 12 16 16
    7 9 7 4 1
    13 4 12 11 5
    11 12 12 9 4


    Please help describe the values by finding a few statistics.

    The sample size, \(n\)

    The sample size is simply a count of how many values are in the sample.

    The sample total, \(\sum x\)

    The sample total is found by adding up all the values. Informally, we write it as \(\sum x\). More formally, we write it as \(\sum\limits_{i=1}^n x_i\). Notice, both methods refer to the summation operator.

    The sample mean, \(\bar{x}\)

    The sample mean, or in common parlance “the sample average”, is found by dividing the sample total by the sample size. The symbol \(\bar{x}\) is pronounced “x bar”. Please round to the nearest tenth. \[\bar{x} = \frac{\sum x}{n}\]

    The sample median

    The median represents a boundary between the lower half of the values and the upper half of the values. This should be review. Remember, it helps to sort the data. We could write a formula using a sort function and floor and ceiling functions:

    \[\mathrm{median}(x) ~=~ \frac{\mathrm{sort}(x)_{\lceil n/2\rceil}+\mathrm{sort}(x)_{\lfloor n/2+1\rfloor}}{2}\]

    But, you’ll probably just want to sort the values. Then, if the sample size is odd, find the middle value. If the sample size is even, find the mean of the two middle values.

    A conditional count

    In data analysis, we often use conditional counts. In this case, I want you to count how many values are at most 7. I am using the # symbol to represent a count, and in the parentheses I am denoting a condition (with an inequality).

    A sample proportion

    What proportion of the values are at least 12? A proportion is a conditional count divided by the sample size. Please round to the nearest hundredth.



    Solution


  30. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A spinner was made to model a discrete-uniform distribution.

    plot of chunk unnamed-chunk-1

    The spinner was spun many times, producing the values below.

    x
    22 10 4 13 13 26 27
    28 2 22 7 29 25 13
    10 27 3 15 19 3 9
    14 7 4 23 19 22 23
    14 26 9 24 5 20 14
    5 29 15 10 11 26 19
    17 23 14 7 9 17 14
    16 12 26 11 28 2 7
    11 1 5 4 21 2 20
    22 8 24 1 19 4 15
    23 28 15 7 16 15 26
    14 19 7 14 17 25 3
    4 11 9 24 6 1 25
    3 16 16 19 12 23 19
    26 19 7 26 16 5 3
    29 14 27 6 23 4 20
    21 29 4 17 29 8 23
    5 13 28 6 8 10 16
    26 3 20 28 22 26 28


    Please help describe the values by finding a few statistics.

    The sample size, \(n\)

    The sample size is simply a count of how many values are in the sample.

    The sample total, \(\sum x\)

    The sample total is found by adding up all the values. Informally, we write it as \(\sum x\). More formally, we write it as \(\sum\limits_{i=1}^n x_i\). Notice, both methods refer to the summation operator.

    The sample mean, \(\bar{x}\)

    The sample mean, or in common parlance “the sample average”, is found by dividing the sample total by the sample size. The symbol \(\bar{x}\) is pronounced “x bar”. Please round to the nearest tenth. \[\bar{x} = \frac{\sum x}{n}\]

    The sample median

    The median represents a boundary between the lower half of the values and the upper half of the values. This should be review. Remember, it helps to sort the data. We could write a formula using a sort function and floor and ceiling functions:

    \[\mathrm{median}(x) ~=~ \frac{\mathrm{sort}(x)_{\lceil n/2\rceil}+\mathrm{sort}(x)_{\lfloor n/2+1\rfloor}}{2}\]

    But, you’ll probably just want to sort the values. Then, if the sample size is odd, find the middle value. If the sample size is even, find the mean of the two middle values.

    A conditional count

    In data analysis, we often use conditional counts. In this case, I want you to count how many values are at most 20. I am using the # symbol to represent a count, and in the parentheses I am denoting a condition (with an inequality).

    A sample proportion

    What proportion of the values are greater than 7? A proportion is a conditional count divided by the sample size. Please round to the nearest hundredth.



    Solution


  31. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A spinner was made to model a continuous triangular distribution. It was spun 50 times; those locations are marked.

    plot of chunk unnamed-chunk-1

    The name “triangular distribution” comes from the probability density curve.

    plot of chunk unnamed-chunk-2

    Notice that because the spinner is continuous, it generates data like 351.8498575 and 341.9542846. Of course, a human would probably round to the nearest tenth or so.

    Actual question

    The 50 spins were used to generate a histogram.

    plot of chunk unnamed-chunk-3



    Solution


  32. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A spinner was made to model a continuous triangular distribution. It was spun 50 times; those locations are marked.

    plot of chunk unnamed-chunk-1

    The name “triangular distribution” comes from the probability density curve.

    plot of chunk unnamed-chunk-2

    Notice that because the spinner is continuous, it generates data like 691.7411754 and 672.0175371. Of course, a human would probably round to the nearest tenth or so.

    Actual question

    The 50 spins were used to generate a histogram.

    plot of chunk unnamed-chunk-3



    Solution


  33. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A spinner was made to model a continuous triangular distribution. It was spun 50 times; those locations are marked.

    plot of chunk unnamed-chunk-1

    The name “triangular distribution” comes from the probability density curve.

    plot of chunk unnamed-chunk-2

    Notice that because the spinner is continuous, it generates data like 373.3602202 and 387.7683341. Of course, a human would probably round to the nearest tenth or so.

    Actual question

    The 50 spins were used to generate a histogram.

    plot of chunk unnamed-chunk-3



    Solution


  34. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A spinner was made to model a continuous triangular distribution. It was spun 50 times; those locations are marked.

    plot of chunk unnamed-chunk-1

    The name “triangular distribution” comes from the probability density curve.

    plot of chunk unnamed-chunk-2

    Notice that because the spinner is continuous, it generates data like 792.4623401 and 787.9583191. Of course, a human would probably round to the nearest tenth or so.

    Actual question

    The 50 spins were used to generate a histogram.

    plot of chunk unnamed-chunk-3



    Solution


  35. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A spinner was made to model a continuous triangular distribution. It was spun 50 times; those locations are marked.

    plot of chunk unnamed-chunk-1

    The name “triangular distribution” comes from the probability density curve.

    plot of chunk unnamed-chunk-2

    Notice that because the spinner is continuous, it generates data like 357.502555 and 343.4080951. Of course, a human would probably round to the nearest tenth or so.

    Actual question

    The 50 spins were used to generate a histogram.

    plot of chunk unnamed-chunk-3



    Solution


  36. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A sample of size \(n=200\) was taken from an unknown population. The sample is shown below:

    x
    1 1.03
    2 9.11
    3 9.53
    4 2.96
    5 1.26
    6 1.21
    7 8.60
    8 9.31
    9 1.83
    10 7.93
    11 2.75
    12 4.88
    13 1.81
    14 1.78
    15 1.08
    16 8.04
    17 9.02
    18 8.06
    19 8.30
    20 3.68
    21 7.10
    22 1.97
    23 0.17
    24 8.73
    25 1.35
    26 0.95
    27 1.48
    28 1.63
    29 8.01
    30 0.75
    31 5.39
    32 6.11
    33 8.11
    34 9.09
    35 0.15
    36 0.51
    37 2.40
    38 9.12
    39 3.56
    40 8.60
    41 4.48
    42 0.94
    43 8.58
    44 8.39
    45 9.00
    46 1.53
    47 0.53
    48 3.58
    49 0.99
    50 0.21
    51 1.72
    52 1.25
    53 9.39
    54 9.03
    55 5.08
    56 8.75
    57 0.28
    58 2.03
    59 8.73
    60 1.44
    61 1.81
    62 3.00
    63 9.08
    64 0.02
    65 8.26
    66 9.13
    67 0.07
    68 0.16
    69 9.48
    70 9.21
    71 0.86
    72 0.26
    73 7.87
    74 1.91
    75 0.18
    76 2.31
    77 0.47
    78 6.59
    79 4.16
    80 9.01
    81 1.42
    82 7.67
    83 0.44
    84 9.35
    85 3.42
    86 2.95
    87 1.28
    88 7.77
    89 0.19
    90 9.93
    91 8.33
    92 1.80
    93 4.66
    94 0.40
    95 4.59
    96 6.68
    97 5.86
    98 0.82
    99 9.25
    100 8.79
    101 8.32
    102 2.55
    103 7.38
    104 7.40
    105 7.55
    106 8.51
    107 0.86
    108 0.91
    109 2.85
    110 2.14
    111 8.92
    112 2.79
    113 1.35
    114 8.52
    115 8.98
    116 7.97
    117 8.10
    118 8.94
    119 1.76
    120 1.00
    121 7.19
    122 9.91
    123 9.53
    124 1.59
    125 0.18
    126 7.75
    127 3.86
    128 8.31
    129 6.42
    130 2.20
    131 7.73
    132 1.79
    133 0.37
    134 9.89
    135 8.11
    136 2.38
    137 8.96
    138 7.59
    139 2.70
    140 1.35
    141 0.68
    142 7.08
    143 8.90
    144 7.85
    145 9.35
    146 0.74
    147 7.07
    148 1.44
    149 1.10
    150 1.03
    151 9.29
    152 5.48
    153 9.56
    154 0.17
    155 9.85
    156 1.79
    157 8.82
    158 9.44
    159 5.03
    160 3.48
    161 3.01
    162 9.34
    163 0.20
    164 9.75
    165 6.48
    166 6.16
    167 1.45
    168 9.63
    169 1.25
    170 8.08
    171 1.98
    172 0.22
    173 0.78
    174 0.17
    175 1.48
    176 0.55
    177 2.22
    178 8.75
    179 9.62
    180 1.08
    181 1.47
    182 1.63
    183 0.51
    184 1.28
    185 9.04
    186 9.95
    187 8.71
    188 0.62
    189 4.47
    190 4.67
    191 1.95
    192 7.05
    193 1.58
    194 9.19
    195 1.43
    196 7.84
    197 8.08
    198 8.18
    199 7.59
    200 8.34

    Use those data to draw a histogram.

    Identify the correct histogram below and describe the shape. Note, the bucket size is 1.

    plot of chunk unnamed-chunk-1



    Solution


  37. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A sample of size \(n=200\) was taken from an unknown population. The sample is shown below:

    x
    1 9.15
    2 0.33
    3 0.02
    4 7.02
    5 1.53
    6 1.58
    7 4.20
    8 1.51
    9 4.90
    10 1.69
    11 7.63
    12 8.31
    13 0.89
    14 1.53
    15 0.29
    16 1.72
    17 3.68
    18 2.62
    19 1.10
    20 7.17
    21 1.08
    22 1.16
    23 8.85
    24 1.02
    25 9.98
    26 9.15
    27 0.18
    28 7.64
    29 9.33
    30 0.46
    31 9.87
    32 1.82
    33 0.20
    34 1.57
    35 8.41
    36 1.21
    37 8.89
    38 2.71
    39 1.74
    40 1.82
    41 1.78
    42 6.55
    43 1.69
    44 1.95
    45 5.98
    46 1.42
    47 9.61
    48 9.16
    49 1.79
    50 1.93
    51 5.58
    52 9.22
    53 1.31
    54 9.01
    55 5.36
    56 5.70
    57 3.40
    58 9.92
    59 1.52
    60 0.18
    61 9.31
    62 9.60
    63 2.60
    64 8.92
    65 8.79
    66 1.24
    67 4.09
    68 6.98
    69 9.83
    70 1.95
    71 7.31
    72 0.87
    73 4.67
    74 9.03
    75 8.70
    76 7.27
    77 7.00
    78 3.04
    79 8.40
    80 0.08
    81 8.88
    82 9.93
    83 8.31
    84 9.33
    85 1.15
    86 1.38
    87 1.45
    88 0.70
    89 9.93
    90 1.91
    91 9.49
    92 1.73
    93 1.62
    94 0.24
    95 8.80
    96 8.38
    97 8.31
    98 9.45
    99 0.07
    100 0.18
    101 1.73
    102 7.59
    103 8.23
    104 2.19
    105 9.64
    106 8.82
    107 1.03
    108 9.09
    109 7.11
    110 8.77
    111 0.29
    112 7.67
    113 9.05
    114 1.92
    115 6.46
    116 1.55
    117 9.51
    118 1.97
    119 1.25
    120 7.06
    121 1.18
    122 0.94
    123 9.86
    124 2.96
    125 7.70
    126 0.54
    127 1.76
    128 6.62
    129 8.44
    130 0.98
    131 2.73
    132 0.79
    133 3.70
    134 4.50
    135 9.30
    136 8.96
    137 9.77
    138 1.39
    139 1.95
    140 7.41
    141 8.47
    142 0.44
    143 8.21
    144 0.48
    145 0.50
    146 8.18
    147 9.83
    148 8.40
    149 5.43
    150 9.72
    151 9.40
    152 1.11
    153 8.83
    154 2.48
    155 1.24
    156 2.65
    157 0.75
    158 8.49
    159 8.16
    160 1.46
    161 5.42
    162 8.14
    163 1.20
    164 0.45
    165 2.00
    166 1.03
    167 9.64
    168 5.10
    169 6.16
    170 9.33
    171 1.30
    172 4.02
    173 8.35
    174 9.63
    175 8.03
    176 0.09
    177 8.67
    178 9.26
    179 0.72
    180 9.31
    181 8.80
    182 8.91
    183 3.67
    184 8.98
    185 8.27
    186 8.97
    187 1.51
    188 3.77
    189 8.69
    190 9.25
    191 6.87
    192 9.86
    193 0.34
    194 0.26
    195 7.47
    196 1.50
    197 4.74
    198 2.75
    199 0.78
    200 1.97

    Use those data to draw a histogram.

    Identify the correct histogram below and describe the shape. Note, the bucket size is 1.

    plot of chunk unnamed-chunk-1



    Solution


  38. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A sample of size \(n=200\) was taken from an unknown population. The sample is shown below:

    x
    1 7.44
    2 6.42
    3 2.42
    4 4.36
    5 6.72
    6 2.58
    7 7.48
    8 3.22
    9 6.47
    10 7.65
    11 4.35
    12 7.11
    13 4.88
    14 7.21
    15 5.36
    16 3.60
    17 7.19
    18 3.05
    19 4.55
    20 0.93
    21 3.71
    22 4.69
    23 7.60
    24 2.82
    25 6.60
    26 4.26
    27 7.87
    28 1.72
    29 5.55
    30 4.32
    31 8.42
    32 8.13
    33 6.45
    34 5.38
    35 2.80
    36 7.78
    37 4.94
    38 0.40
    39 2.85
    40 4.79
    41 4.61
    42 4.56
    43 4.39
    44 6.25
    45 4.11
    46 5.33
    47 3.85
    48 4.74
    49 5.01
    50 7.76
    51 8.47
    52 6.54
    53 4.49
    54 4.57
    55 7.82
    56 5.46
    57 3.76
    58 5.86
    59 2.36
    60 2.20
    61 2.39
    62 4.24
    63 3.07
    64 3.60
    65 6.73
    66 1.88
    67 6.49
    68 6.57
    69 3.17
    70 5.24
    71 4.41
    72 3.61
    73 3.45
    74 5.80
    75 3.95
    76 7.21
    77 6.08
    78 4.20
    79 6.91
    80 6.27
    81 2.82
    82 5.64
    83 5.33
    84 7.02
    85 4.62
    86 6.16
    87 6.17
    88 4.10
    89 3.41
    90 3.13
    91 7.40
    92 4.47
    93 4.80
    94 6.03
    95 7.61
    96 2.65
    97 8.90
    98 4.79
    99 2.00
    100 6.14
    101 3.85
    102 4.85
    103 6.16
    104 3.85
    105 7.37
    106 5.40
    107 1.50
    108 5.67
    109 6.74
    110 5.87
    111 4.00
    112 4.69
    113 7.03
    114 7.02
    115 6.34
    116 3.90
    117 4.93
    118 2.94
    119 5.44
    120 4.79
    121 3.60
    122 6.34
    123 8.26
    124 4.94
    125 7.81
    126 7.00
    127 6.21
    128 4.57
    129 2.77
    130 7.80
    131 9.14
    132 7.46
    133 2.00
    134 7.54
    135 8.58
    136 8.34
    137 1.22
    138 9.50
    139 3.76
    140 3.70
    141 5.03
    142 3.91
    143 4.04
    144 2.88
    145 6.55
    146 8.91
    147 5.96
    148 4.28
    149 6.54
    150 5.57
    151 6.63
    152 5.55
    153 7.41
    154 6.54
    155 5.16
    156 3.18
    157 6.90
    158 4.98
    159 1.51
    160 6.37
    161 5.86
    162 8.84
    163 8.39
    164 4.04
    165 2.47
    166 2.85
    167 2.00
    168 2.58
    169 6.27
    170 7.66
    171 5.23
    172 1.27
    173 6.48
    174 4.90
    175 6.19
    176 3.34
    177 5.54
    178 3.24
    179 2.01
    180 4.23
    181 5.94
    182 5.07
    183 3.73
    184 5.26
    185 3.91
    186 7.86
    187 6.90
    188 5.70
    189 2.81
    190 7.19
    191 2.34
    192 2.64
    193 7.10
    194 3.88
    195 5.17
    196 2.43
    197 6.62
    198 7.12
    199 4.01
    200 6.42

    Use those data to draw a histogram.

    Identify the correct histogram below and describe the shape. Note, the bucket size is 1.

    plot of chunk unnamed-chunk-1



    Solution


  39. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A sample of size \(n=200\) was taken from an unknown population. The sample is shown below:

    x
    1 1.38
    2 4.61
    3 0.56
    4 0.37
    5 1.86
    6 1.76
    7 1.31
    8 1.20
    9 1.52
    10 1.82
    11 1.85
    12 2.60
    13 0.33
    14 0.12
    15 4.07
    16 1.90
    17 3.80
    18 2.06
    19 4.07
    20 2.76
    21 0.25
    22 2.39
    23 3.27
    24 2.37
    25 1.43
    26 0.98
    27 2.70
    28 3.60
    29 1.16
    30 0.60
    31 0.77
    32 0.61
    33 0.24
    34 1.21
    35 4.88
    36 6.12
    37 0.11
    38 3.19
    39 3.11
    40 0.95
    41 3.09
    42 1.23
    43 1.72
    44 0.17
    45 5.63
    46 1.53
    47 0.77
    48 6.34
    49 5.50
    50 0.24
    51 2.41
    52 1.00
    53 6.92
    54 0.84
    55 1.21
    56 1.43
    57 0.73
    58 3.84
    59 3.91
    60 1.37
    61 0.08
    62 0.93
    63 3.51
    64 5.82
    65 0.52
    66 4.68
    67 3.73
    68 0.66
    69 1.94
    70 4.21
    71 2.02
    72 3.17
    73 2.25
    74 0.95
    75 3.36
    76 0.38
    77 0.59
    78 4.11
    79 5.52
    80 3.79
    81 1.21
    82 3.14
    83 1.49
    84 3.13
    85 1.77
    86 0.80
    87 4.96
    88 4.77
    89 2.33
    90 5.32
    91 1.33
    92 3.66
    93 4.17
    94 4.47
    95 1.63
    96 1.60
    97 0.05
    98 0.22
    99 1.46
    100 4.80
    101 2.62
    102 0.45
    103 0.42
    104 2.38
    105 3.01
    106 1.49
    107 1.72
    108 0.82
    109 4.73
    110 2.77
    111 1.34
    112 0.11
    113 4.38
    114 0.34
    115 0.57
    116 2.55
    117 1.55
    118 5.15
    119 4.95
    120 3.17
    121 0.37
    122 1.27
    123 1.96
    124 3.33
    125 0.98
    126 3.52
    127 2.55
    128 1.70
    129 4.18
    130 3.35
    131 1.50
    132 5.42
    133 2.38
    134 2.14
    135 0.77
    136 2.98
    137 1.54
    138 4.83
    139 2.02
    140 2.99
    141 3.20
    142 1.53
    143 0.62
    144 2.16
    145 1.84
    146 2.92
    147 0.75
    148 0.35
    149 3.78
    150 0.45
    151 3.00
    152 0.16
    153 0.93
    154 0.08
    155 3.73
    156 3.37
    157 0.98
    158 2.30
    159 2.05
    160 0.25
    161 0.94
    162 2.70
    163 1.02
    164 3.41
    165 2.93
    166 1.36
    167 2.25
    168 4.03
    169 1.52
    170 4.13
    171 0.88
    172 1.03
    173 2.53
    174 3.45
    175 1.21
    176 4.37
    177 0.05
    178 1.09
    179 3.19
    180 5.64
    181 2.33
    182 2.09
    183 6.34
    184 0.62
    185 0.54
    186 1.72
    187 7.94
    188 2.45
    189 0.62
    190 1.31
    191 4.48
    192 3.02
    193 5.72
    194 6.14
    195 1.13
    196 0.46
    197 3.93
    198 3.81
    199 5.18
    200 3.93

    Use those data to draw a histogram.

    Identify the correct histogram below and describe the shape. Note, the bucket size is 1.

    plot of chunk unnamed-chunk-1



    Solution


  40. Question

    S-ID.A: Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.

    A sample of size \(n=200\) was taken from an unknown population. The sample is shown below:

    x
    1 8.30
    2 8.12
    3 6.27
    4 6.04
    5 6.97
    6 8.99
    7 6.28
    8 9.48
    9 9.35
    10 9.24
    11 9.40
    12 8.67
    13 8.40
    14 8.86
    15 9.96
    16 7.90
    17 8.95
    18 7.42
    19 9.21
    20 3.31
    21 8.75
    22 5.18
    23 9.77
    24 8.85
    25 5.92
    26 9.46
    27 6.53
    28 2.69
    29 7.35
    30 6.52
    31 7.66
    32 8.34
    33 8.81
    34 8.75
    35 9.73
    36 9.45
    37 9.93
    38 7.32
    39 9.44
    40 4.63
    41 8.01
    42 9.28
    43 9.19
    44 6.70
    45 6.76
    46 7.71
    47 9.03
    48 6.17
    49 7.27
    50 9.29
    51 7.80
    52 7.43
    53 7.06
    54 9.14
    55 5.04
    56 8.57
    57 5.56
    58 7.91
    59 9.39
    60 8.41
    61 6.80
    62 4.11
    63 4.23
    64 7.73
    65 9.61
    66 8.45
    67 6.20
    68 7.75
    69 8.41
    70 8.84
    71 9.53
    72 5.39
    73 8.75
    74 9.93
    75 6.68
    76 7.68
    77 7.85
    78 9.99
    79 8.43
    80 9.14
    81 8.46
    82 6.70
    83 9.28
    84 5.76
    85 6.34
    86 7.86
    87 7.62
    88 6.11
    89 6.17
    90 8.76
    91 8.50
    92 6.52
    93 5.04
    94 8.86
    95 8.20
    96 5.52
    97 8.26
    98 6.95
    99 6.77
    100 7.21
    101 6.30
    102 7.88
    103 4.49
    104 9.45
    105 9.96
    106 7.38
    107 4.30
    108 6.11
    109 4.21
    110 6.34
    111 9.89
    112 7.31
    113 9.98
    114 9.29
    115 5.16
    116 9.15
    117 8.97
    118 8.60
    119 8.74
    120 9.86
    121 6.79
    122 4.92
    123 8.57
    124 7.44
    125 8.24
    126 9.15
    127 5.13
    128 8.28
    129 8.49
    130 9.27
    131 5.95
    132 8.41
    133 5.70
    134 5.07
    135 9.56
    136 9.51
    137 6.78
    138 7.19
    139 7.41
    140 8.47
    141 9.72
    142 8.00
    143 8.48
    144 5.60
    145 5.32
    146 8.54
    147 8.22
    148 3.68
    149 9.82
    150 5.65
    151 8.52
    152 8.96
    153 8.35
    154 5.31
    155 7.61
    156 7.77
    157 7.71
    158 8.74
    159 8.09
    160 9.36
    161 7.47
    162 8.19
    163 9.79
    164 9.31
    165 8.82
    166 6.40
    167 6.66
    168 5.72
    169 5.28
    170 8.82
    171 3.34
    172 9.67
    173 5.22
    174 9.26
    175 6.16
    176 8.50
    177 6.93
    178 5.65
    179 9.19
    180 8.93
    181 7.83
    182 9.39
    183 9.79
    184 7.09
    185 9.48
    186 8.32
    187 9.88
    188 8.13
    189 9.04
    190 8.02
    191 5.05
    192 9.31
    193 6.16
    194 7.24
    195 7.99
    196 3.56
    197 9.03
    198 6.96
    199 9.68
    200 5.88

    Use those data to draw a histogram.

    Identify the correct histogram below and describe the shape. Note, the bucket size is 1.

    plot of chunk unnamed-chunk-1



    Solution


  41. Question

    S-IC.A.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

    In table-top role-playing games, it is common to use dice notation. As an example, if a spell deals 9d8 damage, the player will roll nine 8-sided dice (where each die follows a discrete uniform distribution between 1 and 8) and add all the pips.

    Please use a spreadsheet to simulate 1000 repetitions of 9d8. From the simulation, determine an interval of typical totals (the boundaries of the middle 95%) and a margin of error (half the width of the interval). You might find this guide helpful.

    Your values might differ slightly from the options; pick the closest.

    Now, as a statistician, you’ve decided that another player’s collection of nine 8-sided dice look suspicious. You decide to run a quick test by rolling them all. You decide ahead of time that if the total is inside the interval you will stay quiet. However, if the total is outside the interval, you will shout “guilty!”

    When you roll them, you get a total of 18. What do you do?



    Solution


  42. Question

    S-IC.A.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

    In table-top role-playing games, it is common to use dice notation. As an example, if a spell deals 15d10 damage, the player will roll fifteen 10-sided dice (where each die follows a discrete uniform distribution between 1 and 10) and add all the pips.

    Please use a spreadsheet to simulate 1000 repetitions of 15d10. From the simulation, determine an interval of typical totals (the boundaries of the middle 95%) and a margin of error (half the width of the interval). You might find this guide helpful.

    Your values might differ slightly from the options; pick the closest.

    Now, as a statistician, you’ve decided that another player’s collection of fifteen 10-sided dice look suspicious. You decide to run a quick test by rolling them all. You decide ahead of time that if the total is inside the interval you will stay quiet. However, if the total is outside the interval, you will shout “guilty!”

    When you roll them, you get a total of 100. What do you do?



    Solution


  43. Question

    S-IC.A.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

    In table-top role-playing games, it is common to use dice notation. As an example, if a spell deals 6d8 damage, the player will roll six 8-sided dice (where each die follows a discrete uniform distribution between 1 and 8) and add all the pips.

    Please use a spreadsheet to simulate 1000 repetitions of 6d8. From the simulation, determine an interval of typical totals (the boundaries of the middle 95%) and a margin of error (half the width of the interval). You might find this guide helpful.

    Your values might differ slightly from the options; pick the closest.

    Now, as a statistician, you’ve decided that another player’s collection of six 8-sided dice look suspicious. You decide to run a quick test by rolling them all. You decide ahead of time that if the total is inside the interval you will stay quiet. However, if the total is outside the interval, you will shout “guilty!”

    When you roll them, you get a total of 46. What do you do?



    Solution


  44. Question

    S-IC.A.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

    In table-top role-playing games, it is common to use dice notation. As an example, if a spell deals 12d8 damage, the player will roll twelve 8-sided dice (where each die follows a discrete uniform distribution between 1 and 8) and add all the pips.

    Please use a spreadsheet to simulate 1000 repetitions of 12d8. From the simulation, determine an interval of typical totals (the boundaries of the middle 95%) and a margin of error (half the width of the interval). You might find this guide helpful.

    Your values might differ slightly from the options; pick the closest.

    Now, as a statistician, you’ve decided that another player’s collection of twelve 8-sided dice look suspicious. You decide to run a quick test by rolling them all. You decide ahead of time that if the total is inside the interval you will stay quiet. However, if the total is outside the interval, you will shout “guilty!”

    When you roll them, you get a total of 33. What do you do?



    Solution


  45. Question

    S-IC.A.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

    In table-top role-playing games, it is common to use dice notation. As an example, if a spell deals 14d6 damage, the player will roll fourteen 6-sided dice (where each die follows a discrete uniform distribution between 1 and 6) and add all the pips.

    Please use a spreadsheet to simulate 1000 repetitions of 14d6. From the simulation, determine an interval of typical totals (the boundaries of the middle 95%) and a margin of error (half the width of the interval). You might find this guide helpful.

    Your values might differ slightly from the options; pick the closest.

    Now, as a statistician, you’ve decided that another player’s collection of fourteen 6-sided dice look suspicious. You decide to run a quick test by rolling them all. You decide ahead of time that if the total is inside the interval you will stay quiet. However, if the total is outside the interval, you will shout “guilty!”

    When you roll them, you get a total of 70. What do you do?



    Solution


  46. Question

    S-IC.A.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

    You have heard from a good authority (education research group) that a few years ago a survey showed that in Berkshire County the proportion of teachers who incorporate technology in their lessons is \(p=0.61\). You think that maybe the population proportion has changed with the times. You decide to test your hunch by getting a simple random sample of the teachers in Berkshire County.

    But, you don’t just start gathering data. You need to have a clear test in mind before gathering any data; otherwise you might change the test in response to the data (see post hoc analysis).

    Decide on test

    First, you decide on a sample size. You carefully balance your expectations of time, cost, and enthusiasm to decide on a sample size of \(n=14\).

    For your test, you decide on a significance level of 5%. You will use a simulation to find an interval of the middle 95% of outcomes. Then,

    Simulation

    With the sample size \(n=14\) in mind you are able to run a (repeated) simulation while assuming \(p=0.61\). To do that in a spreadsheet, each cell will use =IF(RAND()<0.61,1,0), which uses 1 for affirmative and 0 for negative.

    Based on your simulation, what are the most reasonable boundaries of the interval of typical (middle 95%) results?

    Let’s also get a margin of error by halving the difference of those boundaries.

    Collect data

    Phew! You’ve set your study design and you’ve simulated a model. Now you go collect data to see if it behaves like the model. Here are your raw data:

    1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

    What do you conclude?



    Solution


  47. Question

    S-IC.A.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

    You have heard from a good authority (veterinary research) that a few years ago a survey showed that in Berkshire County the proportion of cats that enjoy catnip is \(p=0.82\). You think that maybe the population proportion has changed with the times. You decide to test your hunch by getting a simple random sample of the cats in Berkshire County.

    But, you don’t just start gathering data. You need to have a clear test in mind before gathering any data; otherwise you might change the test in response to the data (see post hoc analysis).

    Decide on test

    First, you decide on a sample size. You carefully balance your expectations of time, cost, and enthusiasm to decide on a sample size of \(n=16\).

    For your test, you decide on a significance level of 5%. You will use a simulation to find an interval of the middle 95% of outcomes. Then,

    Simulation

    With the sample size \(n=16\) in mind you are able to run a (repeated) simulation while assuming \(p=0.82\). To do that in a spreadsheet, each cell will use =IF(RAND()<0.82,1,0), which uses 1 for affirmative and 0 for negative.

    Based on your simulation, what are the most reasonable boundaries of the interval of typical (middle 95%) results?

    Let’s also get a margin of error by halving the difference of those boundaries.

    Collect data

    Phew! You’ve set your study design and you’ve simulated a model. Now you go collect data to see if it behaves like the model. Here are your raw data:

    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0

    What do you conclude?



    Solution


  48. Question

    S-IC.A.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

    You have heard from a good authority (animal advocacy group) that a few years ago a survey showed that in Berkshire County the proportion of people who believe in animal rights is \(p=0.76\). You think that maybe the population proportion has changed with the times. You decide to test your hunch by getting a simple random sample of the people in Berkshire County.

    But, you don’t just start gathering data. You need to have a clear test in mind before gathering any data; otherwise you might change the test in response to the data (see post hoc analysis).

    Decide on test

    First, you decide on a sample size. You carefully balance your expectations of time, cost, and enthusiasm to decide on a sample size of \(n=15\).

    For your test, you decide on a significance level of 5%. You will use a simulation to find an interval of the middle 95% of outcomes. Then,

    Simulation

    With the sample size \(n=15\) in mind you are able to run a (repeated) simulation while assuming \(p=0.76\). To do that in a spreadsheet, each cell will use =IF(RAND()<0.76,1,0), which uses 1 for affirmative and 0 for negative.

    Based on your simulation, what are the most reasonable boundaries of the interval of typical (middle 95%) results?

    Let’s also get a margin of error by halving the difference of those boundaries.

    Collect data

    Phew! You’ve set your study design and you’ve simulated a model. Now you go collect data to see if it behaves like the model. Here are your raw data:

    1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0

    What do you conclude?



    Solution


  49. Question

    S-IC.A.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

    You have heard from a good authority (nutrition association) that a few years ago a survey showed that in Berkshire County the proportion of people who cook healthy meals at home is \(p=0.66\). You think that maybe the population proportion has changed with the times. You decide to test your hunch by getting a simple random sample of the people in Berkshire County.

    But, you don’t just start gathering data. You need to have a clear test in mind before gathering any data; otherwise you might change the test in response to the data (see post hoc analysis).

    Decide on test

    First, you decide on a sample size. You carefully balance your expectations of time, cost, and enthusiasm to decide on a sample size of \(n=24\).

    For your test, you decide on a significance level of 5%. You will use a simulation to find an interval of the middle 95% of outcomes. Then,

    Simulation

    With the sample size \(n=24\) in mind you are able to run a (repeated) simulation while assuming \(p=0.66\). To do that in a spreadsheet, each cell will use =IF(RAND()<0.66,1,0), which uses 1 for affirmative and 0 for negative.

    Based on your simulation, what are the most reasonable boundaries of the interval of typical (middle 95%) results?

    Let’s also get a margin of error by halving the difference of those boundaries.

    Collect data

    Phew! You’ve set your study design and you’ve simulated a model. Now you go collect data to see if it behaves like the model. Here are your raw data:

    1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

    What do you conclude?



    Solution


  50. Question

    S-IC.A.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

    You have heard from a good authority (pet behavior study) that a few years ago a survey showed that in Berkshire County the proportion of dogs that enjoy swimming is \(p=0.58\). You think that maybe the population proportion has changed with the times. You decide to test your hunch by getting a simple random sample of the dogs in Berkshire County.

    But, you don’t just start gathering data. You need to have a clear test in mind before gathering any data; otherwise you might change the test in response to the data (see post hoc analysis).

    Decide on test

    First, you decide on a sample size. You carefully balance your expectations of time, cost, and enthusiasm to decide on a sample size of \(n=15\).

    For your test, you decide on a significance level of 5%. You will use a simulation to find an interval of the middle 95% of outcomes. Then,

    Simulation

    With the sample size \(n=15\) in mind you are able to run a (repeated) simulation while assuming \(p=0.58\). To do that in a spreadsheet, each cell will use =IF(RAND()<0.58,1,0), which uses 1 for affirmative and 0 for negative.

    Based on your simulation, what are the most reasonable boundaries of the interval of typical (middle 95%) results?

    Let’s also get a margin of error by halving the difference of those boundaries.

    Collect data

    Phew! You’ve set your study design and you’ve simulated a model. Now you go collect data to see if it behaves like the model. Here are your raw data:

    1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0

    What do you conclude?



    Solution


  51. Question

    S-IC.B.4: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

    A scientist wants to estimate the average mass of adult lab rats under certain conditions. She randomly selects a sample of \(n=12\) mice from the population, and finds their masses in grams, to the nearest gram.

    505 513 492 453 519 435 450 419 541 464 464 474


    Your goals are:

    Your answers (pick closest to your answer):



    Solution


  52. Question

    S-IC.B.4: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

    A scientist wants to estimate the average mass of adult lab rats under certain conditions. She randomly selects a sample of \(n=12\) mice from the population, and finds their masses in grams, to the nearest gram.

    545 536 524 523 515 564 560 519 549 547 563 533


    Your goals are:

    Your answers (pick closest to your answer):



    Solution


  53. Question

    S-IC.B.4: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

    A scientist wants to estimate the average mass of adult lab rats under certain conditions. She randomly selects a sample of \(n=10\) mice from the population, and finds their masses in grams, to the nearest gram.

    430 437 594 576 570 504 473 497 494 486


    Your goals are:

    Your answers (pick closest to your answer):



    Solution


  54. Question

    S-IC.B.4: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

    A scientist wants to estimate the average mass of adult lab rats under certain conditions. She randomly selects a sample of \(n=11\) mice from the population, and finds their masses in grams, to the nearest gram.

    504 538 455 522 546 555 591 529 605 499 550


    Your goals are:

    Your answers (pick closest to your answer):



    Solution


  55. Question

    S-IC.B.4: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

    A scientist wants to estimate the average mass of adult lab rats under certain conditions. She randomly selects a sample of \(n=8\) mice from the population, and finds their masses in grams, to the nearest gram.

    579 554 541 512 558 563 571 621


    Your goals are:

    Your answers (pick closest to your answer):



    Solution